Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM

The tree-level S-matrix of Einstein’s theory is known to have a representation as an integral over the moduli space of punctured spheres localized to the solutions of the scattering equations. In this paper we introduce three operations that can be applied on the integrand in order to produce other theories. Starting in d + M dimensions we use dimensional reduction to construct Einstein-Maxwell with gauge group U(1) M . The second operation turns gravitons into gluons and we call it “squeezing”. This gives rise to a formula for all multi-trace mixed amplitudes in Einstein-Yang-Mills. Dimensionally reducing Yang-Mills we find the S-matrix of a special Yang-Mills-Scalar (YMS) theory, and by the squeezing operation we find that of a YMS theory with an additional cubic scalar vertex. A corollary of the YMS formula gives one for a single massless scalar with a ϕ 4 interaction. Starting again from Einstein’s theory but in d + d dimensions we introduce a “generalized dimensional reduction” that produces the Born-Infeld theory or a special Galileon theory in d dimensions depending on how it is applied. An extension of Born-Infeld formula leads to one for the Dirac-Born-Infeld (DBI) theory. By applying the same operation to Yang-Mills we obtain the U( N ) non-linear sigma model (NLSM). Finally, we show how the Kawai-Lewellen-Tye relations naturally follow from our formulation and provide additional connections among these theories. One such relation constructs DBI from YMS and NLSM

Scattering of massless particles: scalars, gluons and gravitons

In a recent note we presented a compact formula for the complete tree-level S-matrix of pure Yang-Mills and gravity theories in arbitrary spacetime dimension. In this paper we show that a natural formulation also exists for a massless colored cubic scalar theory. In Yang-Mills, the formula is an integral over the space of n marked points on a sphere and has as integrand two factors. The first factor is a combination of Parke-Taylor-like terms dressed with U( N ) color structures while the second is a Pfaffian. The S-matrix of a U( N ) × U( Ñ ) cubic scalar theory is obtained by simply replacing the Pfaffian with a U( Ñ ) version of the previous U( N ) factor. Given that gravity amplitudes are obtained by replacing the U( N ) factor in Yang-Mills by a second Pfaffian, we are led to a natural color-kinematics correspondence. An expansion of the integrand of the scalar theory leads to sums over trivalent graphs and are directly related to the KLT matrix. Combining this and the Yang-Mills formula we find a connection to the BCJ color-kinematics duality as well as a new proof of the BCJ doubling property that gives rise to gravity amplitudes. We end by considering a special kinematic point where the partial amplitude simply counts the number of color-ordered planar trivalent trees, which equals a Catalan number. The scattering equations simplify dramatically and are equivalent to a special Y-system with solutions related to roots of Chebyshev polynomials. The sum of the integrand over the solutions gives rise to a representation of Catalan numbers in terms of eigenvectors and eigenvalues of the adjacency matrix of an A -type Dynkin diagram

Einstein-Yang-Mills scattering amplitudes from scattering equations

We present the building blocks that can be combined to produce tree-level S-matrix elements of a variety of theories with various spins mixed in arbitrary dimensions. The new formulas for the scattering of n massless particles are given by integrals over the positions of n points on a sphere restricted to satisfy the scattering equations. As applications, we obtain all single-trace amplitudes in Einstein-Yang-Mills (EYM) theory, and generalizations to include scalars. Also in EYM but extended by a B-field and a dilaton, we present all double-trace gluon amplitudes. The building blocks are made of Pfaffians and Parke-Taylor-like factors of subsets of particle labels

Who you gonna call? Runaway ghosts, higher derivatives and time-dependence in EFTs

We briefly review the formulation of effective field theories (EFTs) in timedependent situations, with particular attention paid to their domain of validity. Our main interest is the extent to which solutions of the EFT capture the dynamics of the full theory. For a simple model we show by explicit calculation that the low-energy action obtained from a sensible UV completion need not take the restrictive form required to obtain only secondorder field equations, and we clarify why runaway solutions are nevertheless typically not a problem for the EFT. Although our results will not be surprising to many, to our knowledge they are only mentioned tangentially in the EFT literature, which (with a few exceptions) largely addresses time-independent situations

Soft asymptotics with mass gap

From the operator product expansion the gluon condensate controls a certain power law correction to the ultraviolet behavior of the gauge theory. This is reflected by the asymptotic behavior of the effective gluon mass function as determined by its Schwinger–Dyson equation. We show that the current state of the art determination of the gluon mass function by Binosi, Ibanez and Papavassiliou points to a vanishing gluon condensate. If this is correct then the vacuum energy also vanishes in massless QCD. This result can be interpreted as a statement about a softness in the ultraviolet behavior and the consistency of this behavior with a mass gap

Quantum fields and entanglement on a curved lightfront

We consider field quantization on an arbitrary null hypersurface in curved spacetime. We discuss the de Sitter horizon as the simplest example, relating the horizon quantization to the standard Fock space in the cosmological patch. We stress the universality of null-hypersurface kinematics, using it to generalize the Unruh effect to vacuum or thermal states with respect to null “time translations” on arbitrary (e.g. non-stationary) horizons. Finally, we consider a general pure state on a null hypersurface, which is divided into past and future halves, as when a bifurcation surface divides an event horizon. We present a closed-form recipe for reducing such a pure state into a mixed state on each half-hypersurface. This provides a framework for describing entanglement between spacetime regions directly in terms of their causal horizons. To illustrate our state-reduction recipe, we use it to derive the Unruh effect

Antipodally symmetric gauge fields and higher-spin gravity in de Sitter space

We study gauge fields of arbitrary spin in de Sitter space. These include Yang-Mills fields and gravitons, as well as the higher-spin fields of Vasiliev theory. We focus on antipodally symmetric solutions to the field equations, i.e. ones that live on “elliptic” de Sitter space d S 4 / ℤ 2 $$d{S}_4/{\mathrm{\mathbb{Z}}}_2$$ . For free fields, we find spanning sets of such solutions, including boundary-to-bulk propagators. We find that free solutions on d S 4 / ℤ 2 $$d{S}_4/{\mathrm{\mathbb{Z}}}_2$$ can only have one of the two types of boundary data at infinity, meaning that the boundary 2-point functions vanish. In Vasiliev theory, this property persists order by order in the interaction, i.e. the boundary n -point functions in d S 4 / ℤ 2 $$d{S}_4/{\mathrm{\mathbb{Z}}}_2$$ all vanish. This implies that a higher-spin dS/CFT based on the Lorentzian d S 4 / ℤ 2 $$d{S}_4/{\mathrm{\mathbb{Z}}}_2$$ action is empty. For more general interacting theories, such as ordinary gravity and Yang-Mills, we can use the free-field result to define a well-posed perturbative initial value problem in d S 4 / ℤ 2 $$d{S}_4/{\mathrm{\mathbb{Z}}}_2$$

Emergent spacetime in stochastically evolving dimensions

Changing the dimensionality of the space–time at the smallest and largest distances has manifold theoretical advantages. If the space is lower dimensional in the high energy regime, then there are no ultraviolet divergencies in field theories, it is possible to quantize gravity, and the theory of matter plus gravity is free of divergencies or renormalizable. If the space is higher dimensional at cosmological scales, then some cosmological problems (including the cosmological constant problem) can be attacked from a completely new perspective. In this paper, we construct an explicit model of “evolving dimensions” in which the dimensions open up as the temperature of the universe drops. We adopt the string theory framework in which the dimensions are fields that live on the string worldsheet, and add temperature dependent mass terms for them. At the Big Bang, all the dimensions are very heavy and are not excited. As the universe cools down, dimensions open up one by one. Thus, the dimensionality of the space we live in depends on the energy or temperature that we are probing. In particular, we provide a kinematic Brandenberger–Vafa argument for how a discrete causal set, and eventually a continuum (3+1) -dim spacetime along with Einstein gravity emerges in the Infrared from the worldsheet action. The (3+1) -dim Planck mass and the string scale become directly related, without any compactification. Amongst other predictions, we argue that LHC might be blind to new physics even if it comes at the TeV scale. In contrast, cosmic ray experiments, especially those that can register the very beginning of the shower, and collisions with high multiplicity and density of particles, might be sensitive to the dimensional cross-over

Disorder in the early universe

Little is known about the microscopic physics that gave rise to inflation in our universe. There are many reasons to wonder if the underlying description requires a careful arrangement of ingredients or if inflation was the result of an essentially random process. At a technical level, randomness in the microphysics of inflation is closely related to disorder in solids. We develop the formalism of disorder for inflation and investigate the observational consequences of quenched disorder. We find that a common prediction is the presence of additional noise in the power spectrum or bispectrum. At a phenomenological level, these results can be recast in terms of a modulating field, allowing us to write the quadratic maximum likelihood estimator for this noise. Preliminary constraints on disorder can be derived from existing analyses but significant improvements should be possible with a dedicated treatment

Black holes in modified gravity (MOG)

The field equations for scalar–tensor–vector gravity (STVG) or modified gravity (MOG) have a static, spherically symmetric black hole solution determined by the mass M with two horizons. The strength of the gravitational constant is G=GN(1+α) where α is a parameter. A regular singularity-free MOG solution is derived using a nonlinear field dynamics for the repulsive gravitational field component and a reasonable physical energy-momentum tensor. The Kruskal–Szekeres completion of the MOG black hole solution is obtained. The Kerr-MOG black hole solution is determined by the mass M , the parameter α and the spin angular momentum J=Ma . The equations of motion and the stability condition of a test particle orbiting the MOG black hole are derived, and the radius of the black hole photosphere and the shadows cast by the Schwarzschild-MOG and Kerr-MOG black holes are calculated. A traversable wormhole solution is constructed with a throat stabilized by the repulsive component of the gravitational field